( viii ) Vector Product of Two Unit Vectors If a and b are unit vectors , then a = | a | = 1 , b = | b | = 1 ∴ a * b = ab sin θ n = ( sin theta ;). n
( ix ) Vector Product is not Commutative The two vector products a * b and b * a are equal in magnitude but opposite in direction .
i . e ., b * a = - a * b ……..( i )
( x ) The vector product of a vector a with itself is null vector , i . e ., a * a = 0 .
( xi ) Distributive Law For any three vectors a , b , c a * ( b + c ) = ( a * b ) + ( a * c )
( xii ) Area of a Triangle and Parallelogram
( a ) The vector area of a ΔABC is equal to 1 / 2 | AB * AC | or 1 / 2 | BC * BA | or 1 / 2 | CB *
CA |.
( b ) The area of a ΔABC with vertices having PV ’ s a , b , c respectively , is 1 / 2 | a * b + b * c + c
* a |.
( c ) The points whose PV ’ s are a , b , c are collinear , if and only if a * b + b * c + c * a
( d ) The area of a parallelogram with adjacent sides a and b is | a * b |.
( e ) The area of a Parallelogram with diagonals a and b is 1 / 2 | a * b |.
( f ) The area of a quadrilateral ABCD is equal to 1 / 2 | AC * BD |.